I'm not sure what you mean, buy "sounding like a mic'd classical". That compressed sound is hard to avoid in recordings because the dynamic range is very difficult with classical - you get extreme quietness contrasted with huge bass hits. That being said, nothing sounds better for classical than a mic. Piezo pickups have been pretty horrid at accurate reproduction for me. I never use the one in my Alhambra unless I'm playing it with the rest of the band and I don't care about accuracy.
The sound I am able to get get has to do with:
Mic Placement - I generally place it between the 12th and 15th fret, 5-6" from the frets, slightly off axis, pointing toward the headstock. Off axis is important as well as staying away from direct interaction with the sound hole. If you place it head-on at the sound hole, you can get a lot of proximity effect in the form of major boominess and thump. Off axis also helps to avoid excessive bass response. However, it's secondary to placement.
Mic type - I generally recommend a small diaphragm condenser like a KM184 or SM81. I may sell my KM184 in flavor of a Rode large diaphragm. I'll never sell the SM81. If you only have a limited selection, SM57's or SM58s do OK, but need a lot of amplification to sound good and even then may seem compressed.
Mic Preamp - most people are tied to the mixer. But if you can supply a quality mic preamp, you can help shape the output so that it's more natural. I have one myself, but have never had to use it in our church.
Sound System - I'm not sure what to say here. You have what you have. I know that our system is very good for reproducing classical and it sounds very natural. No compression and no clipping. The signal path is mic -> Yamaha om1v -> Crown Amp -> speaker. Sounds excellent. About 20% of the sound that makes it to the back is the guitar itself, the other half is reinforcement.
System Settings - EQ is important for a given room. Our auditorium tends to be rather tinny. I usually take down the highs to remove sibilance and double check the bass to remove any excessive bass and boom. Depending on the room conditions, (we have an expandable auditorium), I'll add reverb and/or delay. But by far, the hardest thing to get right for me has been a good warm bass that is uncompressed. I think I've done that pretty well.
This has been my experience anyhow.
Yikes! This was a long post. Sorry for that